Create a table for the reflection about the origin of the function

Question

f(x) = (x + 2)} + 1.
Hint: First create a table of values for f (x) evaluated at x = 0, x = -1.,
and x = -2.

Answer 1

To reflect on the function \( f(x) = (x + 2) + 1 \), let's first simplify it:

\[ f(x) = x + 2 + 1 = x + 3 \]

Now, we can create a table of values by evaluating \( f(x) \) at \( x = 0, -1, \text{ and } -2 \).

Here is the table of values for \( f(x) \):

| \( x \) | \( f(x) = x + 3 \) | \( f(x) \) Value | |:---------:|:-------------------:|:-----------------:| | 0 | \( 0 + 3 \) | 3 | | -1 | \( -1 + 3 \) | 2 | | -2 | \( -2 + 3 \) | 1 |

Table of Values:

| \( x \) | \( f(x) \) | |:---------:|:-----------:| | 0 | 3 | | -1 | 2 | | -2 | 1 |

Reflection:

1. Function's properties: The function \( f(x) = x + 3 \) is a linear function, which means it has a constant rate of change and its graph is a straight line.
2. Origin shift: The function is a transformation of the basic linear function \( f(x) = x \) shifted up by 3 units.
3. Values at specified points: The function values show that as \( x \) decreases from 0 to -2, \( f(x) \) also decreases from 3 to 1. This indicates a downward trend in the function as \( x \) moves to the left on the x-axis.
4. Graphical Representation: Plotting these points will show a straight line with a positive slope, indicating that \( f(x) \) increases as \( x \) increases.

If you need any additional analyses or specific reflections about the function, feel free to ask!